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Suppose $\\mathcal{K}$ is an abstract elementary class satisfying the amalgamation and joint embedding properties and that $\\mathcal{K}$ is both $\\mu$- and $\\mu^+$-superstable. If $\\mathcal{K}$ has symmetry for non-$\\mu^+$-splitting, then $\\mathcal{K}$ has symmetry for non-$\\mu$-splitting.\n  This is a new application of towers which were introduced by Shelah and Villaveces \\ci"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1507.01991","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-07-07T23:52:00Z","cross_cats_sorted":[],"title_canon_sha256":"1851e12264fdae3af73be4bf68c1a3995bce9563abb7981eb8091e57313dcda6","abstract_canon_sha256":"22d5dbd34a93c3d217e174a8c56543eb2234e7dd8c4097516ad33df05fe17f1d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:32.925483Z","signature_b64":"uNAoqXUEbkSlzVCkZS/yZ9cnRKju/iJal0EI67D6Q1tYs2rpeyRr9jUw9u7H4RYfnoNMZfTX5iv6gCQHOwblDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9dd27da72426067ad2dfdb32271e7f43064fe538a34d578c166bac2dd74485e3","last_reissued_at":"2026-05-18T01:24:32.924967Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:32.924967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Transferring Symmetry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Monica VanDieren","submitted_at":"2015-07-07T23:52:00Z","abstract_excerpt":"In this paper, we apply results of \\cite{Va3} and use towers to transfer symmetry from $\\mu^+$ down to $\\mu$ in superstable abstract elementary classes without using extra set-theoretic assumptions or tameness.\n  Theorem. 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If $\\mathcal{K}$ has symmetry for non-$\\mu^+$-splitting, then $\\mathcal{K}$ has symmetry for non-$\\mu$-splitting.\n  This is a new application of towers which were introduced by Shelah and Villaveces \\ci"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.01991","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1507.01991","created_at":"2026-05-18T01:24:32.925045+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.01991v2","created_at":"2026-05-18T01:24:32.925045+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.01991","created_at":"2026-05-18T01:24:32.925045+00:00"},{"alias_kind":"pith_short_12","alias_value":"TXJH3JZEEYDH","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"TXJH3JZEEYDHVUW7","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"TXJH3JZE","created_at":"2026-05-18T12:29:42.218222+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TXJH3JZEEYDHVUW73MZCOHT7IM","json":"https://pith.science/pith/TXJH3JZEEYDHVUW73MZCOHT7IM.json","graph_json":"https://pith.science/api/pith-number/TXJH3JZEEYDHVUW73MZCOHT7IM/graph.json","events_json":"https://pith.science/api/pith-number/TXJH3JZEEYDHVUW73MZCOHT7IM/events.json","paper":"https://pith.science/paper/TXJH3JZE"},"agent_actions":{"view_html":"https://pith.science/pith/TXJH3JZEEYDHVUW73MZCOHT7IM","download_json":"https://pith.science/pith/TXJH3JZEEYDHVUW73MZCOHT7IM.json","view_paper":"https://pith.science/paper/TXJH3JZE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.01991&json=true","fetch_graph":"https://pith.science/api/pith-number/TXJH3JZEEYDHVUW73MZCOHT7IM/graph.json","fetch_events":"https://pith.science/api/pith-number/TXJH3JZEEYDHVUW73MZCOHT7IM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TXJH3JZEEYDHVUW73MZCOHT7IM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TXJH3JZEEYDHVUW73MZCOHT7IM/action/storage_attestation","attest_author":"https://pith.science/pith/TXJH3JZEEYDHVUW73MZCOHT7IM/action/author_attestation","sign_citation":"https://pith.science/pith/TXJH3JZEEYDHVUW73MZCOHT7IM/action/citation_signature","submit_replication":"https://pith.science/pith/TXJH3JZEEYDHVUW73MZCOHT7IM/action/replication_record"}},"created_at":"2026-05-18T01:24:32.925045+00:00","updated_at":"2026-05-18T01:24:32.925045+00:00"}